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Recently J. J. Kohn (2005) proved C ∞ hypoellipticity for P k = LL + L|z| 2k L = −L * L − (z k L) * z k L with L = ∂ ∂z + iz ∂ ∂t , (the negative of) a singular sum of squares of complex vector fields on the complex Heisenberg group, an operator which exhibits a loss of k − 1 derivatives. Subsequently, M. Derridj and D. S. Tartakoff proved analytic hypoellipticity for this operator using rather different methods going back to earlier methods of Tartakoff. Those methods also provide an alternate proof of the hypoellip-ticity given by Kohn. In this paper, we consider the equation P m,k = L m L m + L m |z| 2k L m with L m = ∂ ∂z + iz|z| 2m ∂ ∂t , for which the underlying manifold is only of finite type, and prove analytic hypoellipticity using methods of Derridj and Tartakoff. This operator is also subelliptic with large loss of derivatives, but the exact loss plays no role for analytic hypoellipticity. Nonetheless, these methods give a proof of C ∞ hy-poellipticity with precise loss as well, which is to appear in a forthcoming paper by A. Bove, M. Derridj, J. J. Kohn and the author.

We compute explicitly the Bergman and Szego kernels for a class of pseudoconvex domains. The kernels are expressed in terms of Appell's multivariable hypergeometric functions. These explicit formulas are applied to investigate the asymptotic behavior of the Bergman kernel near some weakly pseudoconvex boundary points.

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Abstract.Let (x, r) e Rm x R and u e C2(Rm x R). We discuss local and microlocal analyticity for solutions u to the nonlinear equation ut = f(x, t, u, ux).Here f(x, /, fo . 0 's complex valued and analytic in all arguments. We also assume / to be holomorphic in (Co, C) € C x Cm . In particular we show that WF^ u c Char(/_") where WF^ denotes the analytic wave-front set and Char(L") is the characteristic set of the linearized operatorIf we assume u 6 C3(Äm x R) then we show that the analyticity of u propagates along the elliptic submanifolds of Lu .